A158352 - OEIS

%I #36 Dec 04 2020 19:39:00

%S 2,1,11,14,71,101,111,114,171,1071,1101,1111,1114,1171,11171,14171,

%T 71171,101171,111171,114171,171171,1071171,1101171,1111171,1114171,

%U 1171171,11171171,14171171,71171171,101171171,111171171,114171171,171171171,1071171171,1101171171

%N a(n) is the smallest number whose name in US English contains n vowels.

%C In US English, "101" is written as "one hundred one".

%C From _Michael S. Branicky_, Oct 24 2020 (Start)

%C The sequence counts vowels by counting the instances of the letters 'a', 'e', 'i', 'o', 'u', and 'y' (see Example). If 'y' were not included, then a(5) = 102 ("OnE hUndrEd twO") is the first among subsequent value changes.

%C In extending the sequence to large numbers, the "American system" (Weisstein link), also known as the "short scale" (Wikipedia link), was used. Also, the common written form is adopted ("one thousand one hundred seventeen" not "eleven hundred seventeen"; Wilson link).

%C Also, a(n) = A158353(n) for n >= 3. Proof. For them to differ after (171)^n, which has 9*n vowels, would require either (i) "one [power name] two" be next, preceding "one [power name] one"), which could only occur if [power name] had 9*n-2 vowels; or (ii) [power name] has > 9*n-2 vowels and a subsequent power name has less (within a range depending on n). Neither case occurs for existing names (see Wikipedia Large numbers link). (End)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LargeNumber.html">Large Number</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Names_of_large_numbers">Names of Large Numbers</a>

%H Wiktionary, <a href="https://en.wiktionary.org/wiki/one_hundred_one">one hundred one</a> (US)

%H Wiktionary, <a href="https://en.wiktionary.org/wiki/one_hundred_and_one">one hundred and one</a> (UK)

%H Robert G. Wilson v, <a href="https://oeis.org/A001477/a001477.txt">American English names for the numbers from 0 to 100999 without spaces or hyphens</a>

%e "twO" has 1 vowel, "OnE" has 2, "ElEvEn" has 3, "fOUrtEEn" has 4, "sEvEntY-OnE" has 5, "OnE hUndrEd OnE" has 6, "OnE hUndrEd ElEvEn" has 7.

%t With[{s = Array[StringCount[StringJoin@ Map[If[IntegerQ[#], IntegerName[#], ToString[#]] &, ToExpression@ StringSplit@ IntegerName[#]], {"a", "e", "i", "o", "u", "y"}] &, 10^4]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* _Michael De Vlieger_, Nov 06 2020 *)

%o (Python)

%o from num2words import num2words

%o def A158352(n):

%o   i = 1

%o   while sum(1 for c in num2words(i).replace(' and ', '') if c in "aeiouy")!=n:

%o     i += 1

%o   return i

%o print([A158352(n) for n in range(1,16)]) # _Michael S. Branicky_, Oct 23 2020

%Y Cf. A000052, A079741, A158353.

%K nonn,word

%O 1,1

%A _Rodolfo Kurchan_, Mar 16 2009

%E Edited by _Jon E. Schoenfield_, Sep 29 2018

%E a(9) onwards from _Michael S. Branicky_, Oct 23 2020